Volterra integral equation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Volterra integral equations are a special type of <a href="/facts/Integral_equation/LCu4Imfq">integral equations</a>. They are divided into two groups referred to as the first and the second kind.
A linear Volterra equation of the first kind is

f
        (
        t
        )
        =
        
          ∫
          
            a
          
          
            t
          
        
        K
        (
        t
        ,
        s
        )
        
        x
        (
        s
        )
        
        d
        s
      
    
    {\displaystyle f(t)=\int _{a}^{t}K(t,s)\,x(s)\,ds}

where f is a given function and x is an unknown function to be solved for. A linear Volterra equation of the second kind is

x
        (
        t
        )
        =
        f
        (
        t
        )
        +
        
          ∫
          
            a
          
          
            t
          
        
        K
        (
        t
        ,
        s
        )
        x
        (
        s
        )
        
        d
        s
        .
      
    
    {\displaystyle x(t)=f(t)+\int _{a}^{t}K(t,s)x(s)\,ds.}

In <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>, and in <a href="/facts/Fredholm_theory/q1Kkxd1I">Fredholm theory</a>, the corresponding operators are called <a href="/facts/Volterra_operator/u9Rg1Gk6">Volterra operators</a>. A useful method to solve such equations, the <a href="/facts/Adomian_decomposition_method/hpaxCVBb">Adomian decomposition method</a>, is due to <a href="/facts/George_Adomian/EVXhlECp">George Adomian</a>.
A linear Volterra integral equation is a <a href="/facts/Convolution/PVPrdz9J">convolution</a> equation if

x
        (
        t
        )
        =
        f
        (
        t
        )
        +
        
          ∫
          
            
              t
              
                0
              
            
          
          
            t
          
        
        K
        (
        t
        −
        s
        )
        x
        (
        s
        )
        
        d
        s
        .
      
    
    {\displaystyle x(t)=f(t)+\int _{t_{0}}^{t}K(t-s)x(s)\,ds.}

The function 
 
 
 
 K
 
 
 {\displaystyle K}
 
 in the integral is called the <a href="/facts/Integral_transform/AB2yjHJJ">kernel</a>. Such equations can be analyzed and solved by means of <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> techniques.
For a weakly singular kernel of the form 
 
 
 
 K
 (
 t
 ,
 s
 )
 =
 (
 
 t
 
 2
 
 
 −
 
 s
 
 2
 
 
 
 )
 
 −
 α
 
 
 
 
 {\displaystyle K(t,s)=(t^{2}-s^{2})^{-\alpha }}
 
 with 
 
 
 
 0
 <
 α
 <
 1
 
 
 {\displaystyle 0<\alpha <1}
 
, Volterra integral equation of the first kind can conveniently be transformed into a classical Abel integral equation. 
The Volterra integral equations were introduced by <a href="/facts/Vito_Volterra/gPLlcRu9">Vito Volterra</a> and then studied by <a href="/facts/Traian_Lalescu/5zP0vPku">Traian Lalescu</a> in his 1908 thesis, Sur les équations de Volterra, written under the direction of <a href="/facts/Charles_%C3%89mile_Picard/fq6m32JE">Émile Picard</a>. In 1911, Lalescu wrote the first book ever on integral equations.
Volterra integral equations find application in <a href="/facts/Demography/VktYwWUD">demography</a> as <a href="/facts/Lotka%27s_integral_equation/1SgaXhNy">Lotka's integral equation</a>, the study of <a href="/facts/Viscoelastic/pDhHgx2T">viscoelastic</a> materials,
in <a href="/facts/Actuarial_science/KxmCE0ME">actuarial science</a> through the <a href="/facts/Renewal_equation/1rDc2czB">renewal equation</a>, and in <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a> to describe the flow behavior near finite-sized boundaries.

Volterra integral equation open-in-new

Volterra integral equation